Mathematical Sciences: p-adic Automorphic Forms

本研究涉及p进对称空间的研究，模形式的算术，以及数论和代数几何中的一些计算和算法问题。在p进自同构形式的一般范围内，将解析函数与p进上半平面上测度及其边界测度之间关系的已知结果推广到Drinfeld的高维p进上半空间。这些结果对由德林菲尔德空间均匀化的代数变异体的几何具有启示意义，包括某些志村变异体和德林菲尔德模变异体。它们对一般线性群的p进表示理论也有重要的影响。在模形式领域，首席研究员正在继续研究与“例外零猜想”有关的问题。最后，对Picard-Fuchs微分方程计算算法和Gauss-Manin连接的早期工作进行了扩展。本课题的研究范围是算术几何和自同构形式的一般领域。代数几何是现代数学中最古老的部分之一。在过去的十年里，它已经发展到解决了几个世纪以来一直存在的问题的地步。最初，它用最简单的方程即多项式来处理平面上的图形。今天，该领域不仅利用代数方法，还利用分析和拓扑学方法；相反，它在这些领域被广泛使用。此外，它已被证明在物理学、理论计算机科学、密码学、编码理论和机器人等多种领域都很有用。自同构形式产生于19世纪中期的非欧几里得几何。因此，数学家和物理学家很早就认识到，许多具有根本重要性的物体在其基本性质上是非欧几里得的。这个领域主要关注关于整数的问题，但在使用几何和分析时，它保留了与历史根源的联系，因此与理论物理中的规范理论和信息论中的编码理论等领域的问题联系在一起。本研究涉及p进对称空间的研究，模形式的算术，以及数论和代数几何中的一些计算和算法问题。在p进自同构形式的一般范围内，将解析函数与p进上半平面上测度及其边界测度之间关系的已知结果推广到Drinfeld的高维p进上半空间。这些结果对由德林菲尔德空间统一化的代数变体（包括特定的志村变体和德林菲尔德模变体）的几何意义没有影响。它们对一般线性群的p进表示理论也有重要的影响。在模形式领域，首席研究员正在继续研究与“例外零猜想”有关的问题。最后，对Picard-Fuchs微分方程计算算法和Gauss-Manin连接的早期工作进行了扩展。本课题的研究范围是算术几何和自同构形式的一般领域。代数几何是现代数学中最古老的部分之一。在过去的十年里，它已经发展到解决了几个世纪以来一直存在的问题的地步。最初，它用最简单的方程即多项式来处理平面上的图形。今天，该领域不仅利用代数方法，还利用分析和拓扑学方法；相反，它在这些领域被广泛使用。此外，它已被证明在物理学、理论计算机科学、密码学、编码理论和机器人等多种领域都很有用。自同构形式产生于19世纪中期的非欧几里得几何。因此，数学家和物理学家很早就认识到，许多具有根本重要性的物体在其基本性质上是非欧几里得的。这个领域主要关注关于整数的问题，但在使用几何和分析时，它保留了与历史根源的联系，因此与理论物理中的规范理论和信息论中的编码理论等领域的问题联系在一起。

This research involves the study of p-adic symmetric spaces, the arithmetic of modular forms, and a number of computational and algorithmic questions in number theory and algebraic geometry. In the general area of p-adic automorphic forms, known results about the relationship between analytic functions and measures on the p-adic upper half plane and measures on its boundary are generalized to Drinfeld's higher dimensional p-adic upper half spaces. Such results have implications for the geometry of algebraic varieties which are uniformized by the Drinfeld spaces, including certain Shimura varieties and Drinfeld modular varieties. They also have important consequences for the p-adic representation theory of the general linear group. In the area of modular forms, the principal investigator is continuing to study questions relating to the "exceptional zero conjecture". Finally, earlier work on algorithms for computing Picard-Fuchs differential equations and the Gauss-Manin connection are extended. The research in this project lies in the general areas of arithmetic geometry and automorphic forms. Algebraic geometry is one of the oldest parts of modern mathematics. In the past ten years, it has blossomed to the point where it has solved problems that have stood for centuries. Originally, it treated figures in the plane defined by the simplest of equations, namely polynomials. Today, the field utilizes methods not only from algebra, but also from analysis and topology; conversely, it is extensively used in those fields. Moreover, it has proved itself useful in fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics. Automorphic forms arose out of non-Euclidean geometry in the middle of the nineteenth century. Both mathematicians and physicists have thus long realized that many objects of fundamental importance are non-Euclidean in their basic nature. This field is principally concerned with questions about the whole numbe rs, but in its use of geometry and analysis, it retains connection to its historical roots and thus to problems in areas as diverse as gauge theory in theoretical physics and coding theory in information theory. This research involves the study of p-adic symmetric spaces, the arithmetic of modular forms, and a number of computational and algorithmic questions in number theory and algebraic geometry. In the general area of p-adic automorphic forms, known results about the relationship between analytic functions and measures on the p-adic upper half plane and measures on its boundary are generalized to Drinfeld's higher dimensional p-adic upper half spaces. Such results hate implications for the geometry of algebraic varieties which are uniformized by the Drinfeld spaces, including certain Shimura varieties and Drinfeld modular varieties. They also have important consequences for the p-adic representation theory of the general linear group. In the area of modular forms, the principal investigator is continuing to study questions relating to the "exceptional zero conjecture". Finally, earlier work on algorithms for computing Picard-Fuchs differential equations and the Gauss-Manin connection are extended. The research in this project lies in the general areas of arithmetic geometry and automorphic forms. Algebraic geometry is one of the oldest parts of modern mathematics. In the past ten years, it has blossomed to the point where it has solved problems that have stood for centuries. Originally, it treated figures in the plane defined by the simplest of equations, namely polynomials. Today, the field utilizes methods not only from algebra, but also from analysis and topology; conversely, it is extensively used in those fields. Moreover, it has proved itself useful in fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics. Automorphic forms arose out of non-Euclidean geometry in the middle of the nineteenth century. B oth mathematicians and physicists have thus long realized that many objects of fundamental importance are non-Euclidean in their basic nature. This field is principally concerned with questions about the whole numbers, but in its use of geometry and analysis, it retains connection to its historical roots and thus to problems in areas as diverse as gauge theory in theoretical physics and coding theory in information theory.